Infinite Prandtl Number Convection

We prove an inequality of the type NCR ^1/3(1+log₊ R)^2/3 for the Nusselt number N in terms of the Rayleigh number R for the equations describing three-dimensional Rayleigh–Bénard convection in the limit of infinite Prandtl number.     

ArbitraryN-vortex solutions to the first order Ginzburg-Landau equations

We prove that a set ofN not necessarily distinct points in the plane determine a unique, real analytic solution to the first order Ginzburg-Landau equations with vortex numberN. This solution has the property that the Higgs field vanishes only at the points in the set and the order of vanishing at a given point is determined by the multiplicity of that point in the set. We prove further that these are the onlyC solutions to the first order Ginzburg-Landau equations.This work is supported in part through funds provided under Contract PHY 77-18762     

Deformations of the embedded Einstein spaces

We discuss a method of studying the stability of solutions of Einstein's equations, which can be outlined as follows: Consider an embedding of a given Einstein spaceV ₄ into a pseudo-Euclidean spaceE _p,q ^N (N > 4,p + q =N) (p,q) describing the signature of the spaceE _p,q ^N . Then all the geometrical objects ofV ₄ can be expressed in terms of the embedding functions,Z ^A (x ⁱ ),A = 1, 2,...,N, i = 0, 1, 2, 3. Then let us deform the embedding:Z ^A Z ^A + ^A , being an infinitesimal parameter. The Einstein equations can be developed then in the powers of; we study the equations arising by requirement of the vanishing of the first- or second-order terms. Some partial results concerning the de Sitter, Einstein, and Minkowskian spaces are given.     

A multidimensional radiation-filled universe

We consider a radiation-filled universe which possesses the product symmetry: (N-dimensional space of constant curvature) × (n sphere). The solutions of all the types, within this class, to the classical field equations are given. In the case of theN-dimensional space of zero or negative curvature constant, the solutions exhibit a tendency to approach asymptotically the Kasner-like state in which theN-dimensional subspace expands while then sphere shrinks to the final singularity. Our conclusions based on the phase-diagram method are in agreement with the results concerning the ^N × S ⁿ universe calculated by Sahdev with the help of numerical methods.     

Bäcklund Transformation for Supersymmetric Self-Dual Theories for Semisimple Gauge Groups and a Hierarchy of Ai Solutions

We present a Bdcklund transformation (a discrete symmetry transformation) for the self-duality equations for supersymmetric gauge theories in N-extended super-Minkowski space ^4|4N for an arbitrary semisimple gauge group. For the case of anA ₁ gauge algebra we integrate the transformation starting with a given solution and iterating the process we construct a hierarchy of explicit solutions.     

On the Stability of the O(N)-Invariant and the Cubic-Invariant Three-Dimensional N-Component Renormalization-Group Fixed Points in the Hierarchical Approximation

We compute renormalization-group fixed points and their spectrum in an ultralocal approximation. We study a case of two competing nontrivial fixed points for a three-dimensional real N-component field: the O(N)-invariant fixed point vs. the cubic-invariant fixed point. We compute the critical value N _c of the cubic ⁴-perturbation at the O(N)-fixed point. The O(N)-fixed point is stable under a cubic ⁴-perturbation below N _c; above N _c it is unstable. The Critical value comes out as 2.219435<N _c<2.219436 in the ultralocal approximation. We also compute the critical value of N at the cubic invariant fixed point. Within the accuracy of our computations, the two values coincide.     

Center-of-mass motion of a system of relativistic Dirac particles

The relativistic center-of-mass motion for a system ofN fermions can be exactly separated because of the linearity of the Dirac operators in momenta which is not possible for quadratic Klein-Gordon particles. The covariant equations derived from Maxwell-Dirac field theory are considered. The center-of-mass equation is still a 4 ^N -component spinor equation. We solve these equations for two- and three-body systems, as well as the relative motion for the non-interacting case, and discuss the quantum numbers and identification of eigenstates and eigenvalues. The results apply for both bound and scattering states. Dedicated to the Third Centenary of the Publication of Principia: Corollary IV.... and therefore the common center of gravity of all bodies acting upon each other (excluding external actions and impediments) is either at rest, or moves uniformly in a right line. Is. Newton, Philosophiae Naturalis Principia Mathematica (S. Pepys, Julii 5, 1686, Londini)     

The Camassa-Holm Hierarchy, N-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in: 1. a new Neumann-like N-dimensional system when it is restricted into a symplectic submanifold of ^2N which is proven to be integrable by using the Dirac-Poisson bracket and the r-matrix process; and 2. a new Bargmann-like N-dimensional system when it is considered in the whole ^2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process.     

Integrable nonlinear Klein-Gordon equations and Toda lattices

We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN2 we present a system of (N–1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Bäcklund transformation and superposition formula for the general system.     

Gravity in hyperspin manifolds

We seek the dynamics of a Bergmann manifold: a manifold of dimensionn=N ² supporting a bundle of spinor spaces of dimensionN, and a map from the tangent spaces to the Hermitian spinor forms. Even though the spin-vector is the fundamental variable of the theory, every invariant analytic function depending on and its firstm derivatives alone can be expressed in terms of the chronometric tensorg and its firstm derivatives. Bergmann manifolds of dimensionn > 4 do not have invariant second-order equations for. We find a family of invariant actions which lead tonth-order quasilinear equations of motion on Bergmann manifolds and reduce to the Einstein-Hilbert action forn=2. The resulting gauge particles have spin, 1/2,1, 3/2, and 2.     

Ground-state structure in a highly disordered spin-glass model

We propose a new Ising spin-glass model on Z ^d of Edwards-Anderson type, but with highly disordered coupling magnitudes, in which a greedy algorithm for producing ground states is exact. We find that the procedure for determining (infinite-volume) ground states for this model can be related to invasion percolation with the number of ground states identified as 2 ^N , whereN=N(d) is the number of distinct global components in the invasion forest. We prove thatN(d)= if the invasion connectivity function is square summable. We argue that the critical dimension separatingN=1 andN= isd _c=8. WhenN(d)=, we consider free or periodic boundary conditions on cubes of side lengthL and show that frustration leads to chaoticL dependence withall pairs of ground states occurring as subsequence limits. We briefly discuss applications of our results to random walk problems on rugged landscapes.     

Entropic repulsion of the lattice free field,II. The 0-boundary case

This paper is a continuation of [5]. We consider the Euclidean massless free field on a boxV _N of volumeN ^d with O-boundary condition; that is the centered Gaussian field with covariances given by the Green function of the simple random walk on ^d ,d3, killed as it exitsV _N . We show that the probability, that all the spins are positive in the boxV _N decays exponentially at a surface rateN ^d–1 . This is in contrast with the rateN ^d–2 logN for the infinite field of [5].     

Deformed harmonic oscillators: Coherent states and Bargmann representations

Generalizing the case of the usual harmonic oscillator, we look for Bargmann representations corresponding to deformed harmonic oscillators. Deformed harmonic oscillator algebras are generated by four operators a, a , N and the unity 1 such as [a, N] = a, [a , N] = –a , a a = (N) and aa = (N + 1). We discuss the conditions of existence of a scalar product expressed with a true integral on the space spanned by the e igenstates of a (or a ). We give various examples, in particular we consider functions that are linear combinations of q ^N, q ^–N and unity and that correspond to q-oscillators with Fock-representations or with non-Fock-representations.     

Hyper-Kähler metrics and a generalization of the Bogomolny equations

We generalize the Bogomolny equations to field equations on ^{3 n} and describe a twistor correspondence. We consider a general hyper-Kähler metric in dimension 4n with an action of the torusT ⁿ compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of theT ⁿ -solution of the field equations coming from the twistor space of the metric.     

CyclicL-operator related with a 3-stateR-matrix

We consider the problem of constructing a cyclicL-operator associated with a 3-stateR-matrix related to theU _q (sl(3)) algebra atq ^N =1. This problem is reduced to the construction of a cyclic (i.e. with no highest weight vector) representation of some twelve generating element algebra, which generalizes theU _q (sl(3)) algebra. We found such representation acting inC ^N C ^N C ^N . The necessary conditions of the existence of the intertwining operator for two representations are also discussed.     

Deformed Bosons: Combinatorics of Normal Ordering

We solve the normal ordering problem for (A A) ⁿ where A and A are one mode deformed ([A,A ] = [N+1] – [N]) bosonic ladder operators. The solution generalizes results known for canonical bosons. It involves combinatorial polynomials in the number operator N for which the generating function and explicit expressions are found. Simple deformations provide examples of the method.     

Factorisations for partition functions of random Hermitian matrix models

The partition functionZ _N , for Hermitian-complex matrix models can be expressed as an explicit integral over ^N , whereN is a positive integer. Such an integral also occurs in connexion with random surfaces and models of two dimensional quantum gravity. We show thatZ _N can be expressed as the product of two partition functions, evaluated at translated arguments, for another model, giving an explicit connexion between the two models. We also give an alternative computation of the partition function for the ⁴-model. The approach is an algebraic one and holds for the functions regarded as formal power series in the appropriate ring.     

Non-linear multi-plane wave solutions of self-dual Yang-Mills theory

New solutions of self-dual Yang-Mills (SDYM) equations are constructed in Minkowski space-time for the gauge groupSL(2, ). After proposing a Lorentz covariant formulation of Yang's equations, a set of Ansätze for exact non-linear multiplane wave solutions are proposed. The gauge fields are rational functions ofe ^x·ki (K _i ² =0, 1iN) for these Ansätze. At least, three families of multisoliton type solutions are derived explicitly. Their asymptotic behaviour shows that non-linear waves scatter non-trivially in Minkowski SDYM.On leave from LPTHE Université Paris VI, 4, Place Jussieu, Tour 16, ler étage, F-75230 Paris Cedex 05, France     

Reduction method ofN-body equations

N-body equations of identical-particle systems are explicitly evaluated forN=3 toN=6. The formalism used is of the Faddeev-Yakubovsky type as derived from the so-called chain-of-partition-labeled approach by Cattapan and Vanzani. We compare the resulting equations with the ones from the multi-three cluster coupling model as given by Sawada et al. and establish an important relation between these two classes ofN-body equations.Dedicated to Profs. Erich Schmid and Ivo laus on the occasion of their 60th birthdays     

Instantons of two-dimensional fermionic effective actions by inverse scattering transformation

Effective actions, containing the logarithm of a functional Dirac determinant, appear in 1/N expansions of fermionic theories (N being the number of flavours). We introduce a method to find symmetric solutions of the corresponding non-linear and non-local saddle-point equations. This method consists in using the scattering data of the rotationally symmetric Dirac equation in two dimensions with the angular mometum as a spectral parameter. We apply the method to fermionic theories with scalar and pseudoscalar quartic couplings. The effective action that generates the 1/N expansion admits a closed form in terms of the scattering data only in the particular case when the model is integrable (Gross-Neveu and Chiral Gross-Neveu model). No instanton solutions are present in these two particular cases. This fact, together with the exact results for theS-matrix and form factors, suggests that the 1/N expansion could be convergent. In the general case, the quantum model has an additional dimensionless parameterg _R·g _R± gives the Chiral Gross-Neveu model. Wheng _R>0, tachyons appear. Forg _R0^–, andg _R–, generically complex-action instantons exists, indicating a possibly Borel-summable 1/N expansion.Laboratoire associé au CNRS LA280